Please forgive me for asking if you are a "flat earth truther". We sometimes get those numpties passing by trying to justify their theories (especially the one about Antarctica being a ringed ice wall with the military guarding it to make sure no-one learns the secret).
There have been many flights over the geographic South Pole, the first documented one being in 1935 by Lincoln Ellsworth and Herbert Hollick-Kenyon. See also the first flight over Antarctica.
Pan Am flight 50, using a Boeing 747SP, also flew over both poles.
No scheduled commercial flights over fly the South Pole since there are no great circle routes in use which do so.
Flights from New Zealand to South Africa would fly over Antarctica but no airline currently flies that route.
As to problems with the compass, modern aircraft have no problems since the primary navigation source is GPS. If you were using a compass only, as you got close to the magnetic South pole (remember, this is not even in Antarctica), the compass would become increasingly unreliable. Simply maintain a straight course though, over the pole, and the compass will become increasingly reliable. However, navigating over the magnetic poles using only a compass is the preserve of explorers and the fool-hardy.
This document contains more technical detail regarding navigation. It talks mainly about the North magnetic pole but is applicable to both.
A couple of other considerations when asking about commercial flights is the need for ETOPS certification and the need to carry special survival equipment when flying south of latitude 72 degrees which is very considerable and would require the removal of revenue seats to carry it.
There is a big difference between the nearest suitable diversion airport flying over the Arctic and flying over the Antarctic.
Check the posting for links to the data listed.
Few airlines fly between cities having a great circle route over Antarctica. Nonstop flights between South Africa and New Zealand, or between Perth, Australia and certain destinations in South America (including Buenos Aires and São Paulo), would overfly Antarctica, but no airline has scheduled such flights. Flights between Australia and South America and between Australia and South Africa pass near the Antarctic coastline. Depending on the winds, the Qantas flight QF 63 from Sydney to Johannesburg-O. R. Tambo sometimes flies over the Antarctic Circle to latitude 71 degrees as well and allowing views of the icecap. Qantas also flies nonstop between Sydney and Santiago de Chile, the most southerly polar route. Depending on winds, this flight may reach 55 degrees south latitude, but other times 71 degrees, which is enough to cross the polar ice cap. The polar route across the remote southern Pacific Ocean between South America and Oceania was pioneered by Aerolineas Argentinas, which began service between Buenos Aires via Rio Gallegos to Auckland in the 1980s with a Boeing 747-200 aircraft. Aerolineas Argentinas later operated to Sydney, but ended its flights to New Zealand and Australia in 2014.
Because of ETOPS limitations on twin-engined aircraft—the maximum distance the aircraft can operate from an airport for emergency landings—only 4-engined aircraft such as the Boeing 747 or Airbus A340 can operate routes near Antarctica. Twin-engined aircraft must fly further north, closer to potential diversion airports; for example, when Virgin Australia operated a flight from Melbourne to Johannesburg on twin-engined Boeing 777 aircraft with a 180-minute ETOPS rating, the flight was two hours longer than a Qantas flight from Sydney to Johannesburg. Air New Zealand flies nonstop between Auckland and Buenos Aires-Ezeiza; in 2015, government regulators approved its twin-engined Boeing 777-200ER aircraft that operate the route for a 330-minute ETOPS rating (i.e. its 777 aircraft can fly a maximum 330 minutes away from the nearest diversion airport), an increase from its previous 240-minute ETOPS rating. LATAM Airlines began a nonstop flight between Santiago de Chile and Sydney via Auckland in April 2015 with twin-engined Boeing 787 aircraft with a 330-minute ETOPS rating. LATAM has announced a nonstop flight between Santiago de Chile and Melbourne to begin in October 2017.